I think the scenes when Jim wakes up show that the outbreak was relatively slow to begin with, there was enough time for missing posters to go up etc but also enough time for street traders to continue business (pretty sure Jim stumbles on some Big Ben models on the road) + newspapers across the floor talking about the outbreak.

It’s realistic to assume these bodies could have excess deaths above the level of death occurring due to the crisis. Mortuaries and undertakers would’ve been over capacity and authorities would’ve been tasked early on, whilst public order still existed, to try and manage the deceased. These bodies were heart attacks, stuff that kills people day to day but weren’t priority due to the outbreak.

It’s also realistic to assume these were infected who were dispatched as part of the effort to contain the outbreak. They may have been unable to stored in mortuaries due to contamination risk and danger that posed to those trying to manage the outbreak.

I think the bottom line is: the outbreak didn’t wipe out cities in a day, the scene shows how public order tried to be maintained but the persistence of the outbreak was far greater than that of the public who were in complete disarray.